This chapter presents the results obtained from the experiments which were run and offers an analysis for what the data shows. It breaks down the values for each of the different experiments separated into groups which show similar trends or patterns. An analysis of the experiments which were carried out is also offered to access what validity they hold or whether any improvements or other experiments should also be carried out.

\section{Replication of Ruxton and Saravia with added Gillespie Simulator}
\label{sec:anal.rux}

\subsubsection*{Extinction Medians}
\label{sec:anal.rux.ext}

\begin{figure}[H]
\centering
\begin{tabular}{|c|c|} \hline
\textbf{Model} & \textbf{Extinction Median}\\ \hline
SdF1 & 110.86\\ \hline
SdF2 & 26.01\\ \hline
SdC1 & 20.72\\ \hline
SdC2 & 22.76\\ \hline
RdF1 & 59.07\\ \hline
RdF2 & 26.13\\ \hline
RsR1 & 109.08\\ \hline
RsR2 & 170.6\\ \hline
Gil & -\\ \hline
\end{tabular}
\caption{Extinction Medians averaged over 10,000 runs}
\label{fig:RuxExtMed}
\end{figure}

The extinction medians found for the different update rules from my experiments (see Figure \ref{fig:RuxExtMed}) show some similarities to those found by Ruxton and Saravia, however there are also many discrepancies. It was found in their experiment that the rule RsR1 had the highest extinction median, with a value of 210. This was almost twice as high as the value I found and my results also showed that the rule RsR2 had the highest extinction median and not RsR1. Ruxton and Saravia also found that the RsR rules for time type 1 and 2 were similar (within 13 steps) whereas my results show a much greater difference and that in fact RsR1 and SdF1 have a much closer correlation (difference of only 0.38 steps). All other rules for my experiments were found to be within 4 steps of Ruxton and Saravia's.

No figures are included for the Gillespie simulator as the system did not ever go extinct within 2000 steps for any of the 10,000 runs. When run, the population size for the Gillespie Simulator fluctuated around 8000 showing that systems run this way are considerably more resilient than those run with a discrete step cellular automaton.

\subsubsection*{Persistent Systems}
\label{sec:anal.rux.pers}

\begin{figure}[H]
\centering
\begin{tabular}{|c|c|c|c|c|} \hline
\textbf{Model} & \textbf{Density} & \textbf{Moran's I} & \textbf{LPI} & \textbf{NP}\\ \hline
SdF1 & 0.6074 & 0.118 & 0.607 & 3.9\\ \hline
SdF2 & 0.2798 & 0.217 & 0.034 & 147.9\\ \hline
SdC1 & 0.1372 & 0.260 & 0.007 & 176.5\\ \hline
SdC2 & 0.0096 & 0.000 & 0.002 & 48.6\\ \hline
RdF1 & 0.5751 & 0.126 & 0.574 & 5.7\\ \hline
RdF2 & 0.2795 & 0.216 & 0.034 & 147.9\\ \hline
RsR1 & 0.4445 & 0.088 & 0.421 & 55.2\\ \hline
RsR2 & 0.4609 & 0.074 & 0.445 & 48.1\\ \hline
Gil & 0.8864 & 0.017 & 0.887 & 1.3\\ \hline
\end{tabular}
\caption{Values for the Persistent Systems and Spatial Indices experiments averaged over 10 runs}
\label{fig:RuxPer}
\end{figure}


\begin{figure}[H]
\begin{center}
\includegraphics[scale = 0.75]{images/Ruxt.png}
\end{center}
\caption{Density against Time for all models}
\label{fig:perstChart}
\end{figure}

The length of the runs for my Persistent Systems experiments were 1500 unlike Ruxton and Saravia who only ran the simulations for 200 steps. This was chosen as later experiments with varying probability sets required more time to stabilise so all experiments  were run with a higher step limit for comparability.

The general pairings found in the plot of the densities of the systems for each update rule (shown in Figure \ref{fig:perstChart}) mainly match those found by Ruxton and Saravia. The only notable difference between the two findings is for the two SdC rules. The data plotted for both SdC1 and SdC2 in Ruxton and Saravia's rules seem to suggest that they started with an initial population which was only half the size than that for all other settings, and they also found that the two rules showed the same behaviour with very similar densities. My results showed the same equilibrium density for the SdC1 rule but the SdC2 simulation would go extinct at around 300 steps unlike the findings of Ruxton and Saravia.

It was also found that the Gillespie simulator produced a system which had a  notably higher equilibrium density than any of the discrete step rules with these probabilities. It stabilised around a population size of 9000 which is considerably higher than the following rule (SdF1) with only around 6000.

As would be expected from the similarities in the plotting of the density over time for each update rule, the average densities (see Figure \ref{fig:RuxPer}) also have the same correlation where all the values except for the SdC2 setting closely match the values found by Ruxton and Saravia. The same correlation was found with the Moran's I values for each of the different update rules except for SdC2 which was close to 0 since the systems would go extinct. The Gillespie simulator has the lowest Moran's I value with the highest population signifying that the empty cells were randomly distributed amongst the grid rather than clumped together bringing down the value. The closest values to the Gillespie simulator are the RsR values for both time type 1 and 2 although both of these have a lower population and a slightly higher Moran's I value signifying that the organisms were in clusters which were spread out quite evenly across the entire grid.

\subsubsection*{Spatial Indices}
\label{sec:anal.rux.spat}

The LPI as defined by Ruxton and Saravia is ``the proportion of the total number of sites occupied by the largest patch"\cite{RuxtonSaravia}. I measured this as:
\[
\frac{\text{size of the largest patch}}{\text{size of the grid}}
\]
It is unclear how Ruxton and Saravia calculated these values as for the SdF1 setting, they show an LPI value of 99.5 meaning that the largest patch consisted of 99.5\% of the grid, however in the previous experiment they showed that the population for this setting fluctuated around 68\% of the grid. This shows that we have not calculated the LPI in the same way. It appears as though they may have taken a measure of the proportion of the total population which is a part of the largest patch, i.e.:
\[
\frac{\text{size of the largest patch}}{\text{size of the population of that species}}
\]

Changing my values to this calculation yields the results:
\begin{figure}[H]
\centering
\begin{tabular}{|c|c|} \hline
\textbf{Model} & \textbf{Proportion of Population}\\ 
 & \textbf{in Largest Patch}\\ \hline
SdF1 & 99.87\\ \hline
SdF2 & 12.29\\ \hline
SdC1 & 5.16\\ \hline
SdC2 & 21.05\\ \hline
RdF1 & 99.76\\ \hline
RdF2 & 12.31\\ \hline
RsR1 & 94.75\\ \hline
RsR2 & 96.45\\ \hline
Gil & 99.99\\ \hline
\end{tabular}
\caption{Proportion of the population which is in the largest patch}
\label{fig:RuxPPLP}
\end{figure}

From these values, most of my results are within 3\% of the quoted figures by Ruxton and Saravia with the exception of SdF2 and SdC2 which showed a 6\% and 17\% difference respectively. The SdC2 value would not be expected to be the same as there appear to be differences with all the values for this setting showing there is an implementation difference between my simulator and Ruxton and Saravia's which is resulting in an observable difference.

The Gillespie simulator shows the highest LPI value which would be expected since it has the highest equilibrium density, although when looking at the proportion of the population which is in the largest patch(Figure \ref{fig:RuxPPLP}) it is only slightly higher than the SdF1 and RdF1 values showing that all three simulations result in systems with similar spatial properties in the sense that they all have one large patch which has almost every organism in it.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Multiple Species and Birth and Death Probabilities}
\label{sec:anal.mult}

\subsubsection*{Extinction Medians}
\label{sec:anal.mult.ext}

\begin{figure}[H]
\centering
\begin{tabular}{|c|c|c|} \hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Extinction Median}} \\ \cline{2-3}
& 1 & 2  \\ \hline
SdF1 & 99.7 & 99.2 \\ \hline
SdF2 & 22.8 & 22.8 \\ \hline
SdC1 & 17.4 & 17.4 \\ \hline
SdC2 & 19.4 & 19.5 \\ \hline
RdF1 & 53.7 & 53.2 \\ \hline
RdF2 & 22.8 & 22.8 \\ \hline
RsR1 & 96.9 & 96.6 \\ \hline
RsR2 & 152.7 & 152.5 \\ \hline
Gil & - & - \\ \hline
\end{tabular}
\caption{Extinction Medians for species 1 and 2 averaged over 10,000 runs}
\label{fig:MultExtMed}
\end{figure}

The extinction medians for each species per update rule (see Figure \ref{fig:MultExtMed}) are all within 0.5 steps of each other showing that they both behave in the exact same way which is what would be expected since they have the same birth and death probabilities. However, in comparison to the extinction medians for the same probability settings except with one species (Figure \ref{fig:RuxExtMed}) each setting was lower by between 3 to 13 steps for the simulations with multiple species. Although the difference is very small between the two simulations, the consistency shows that introducing a second species will reduce the resilience of the original species even if the probabilities remain the same. Again in this case no figures are included for the Gillespie simulator as the system would not go extinct with these values but rather fluctuate around a total population size of about 9000 similar to when there was only one species.


%%%%%%%%%%%%%%%%%%%%
\subsection{Species With The Same Birth and Death Probabilities}
\subsubsection*{Persistent Systems}
\label{sec:anal.mult.pers}

\begin{figure}[t!]
\begin{center}
\subfloat[0.99-0.4 versus 0.99-0.4]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.3123 & 0.2985 & 0.018 & 0.31 & 134.5 \\ \hline
SdC1 & 0.0742 & 0.0625 & 0.252 & 0.01 & 190.4 \\ \hline
RdF2 & 0.1494 & 0.1326 & 0.185 & 0.02 & 233.3 \\ \hline
RsR2 & 0.2008 & 0.2614 & -0.002 & 0.11 & 314.3 \\ \hline
Gil & 0.4988 & 0.3878 & -0.169 & 0.49 & 124.8 \\ \hline
\end{tabular}}
\qquad
\subfloat[0.25-0.1 versus 0.25-0.1]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.2855 & 0.2881 & -0.018 & 0.19 & 232.6 \\ \hline
SdC1 & 0.2491 & 0.2609 & 0.010 & 0.12 & 267.1 \\ \hline
RdF2 & 0.2798 & 0.2617 & -0.003 & 0.14 & 250.7 \\ \hline
RsR2 & 0.2762 & 0.2673 & -0.025 & 0.14 & 270.7 \\ \hline
Gil & 0.3100 & 0.2917 & -0.048 & 0.20 & 237.7 \\ \hline
\end{tabular}}

\subfloat[0.0625-0.025 versus 0.0625-0.025]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.2686 & 0.2840 & -0.035 & 0.11 & 302.7 \\ \hline
SdC1 & 0.2721 & 0.2648 & -0.028 & 0.09 & 315.1 \\ \hline
RdF2 & 0.2720 & 0.2727 & -0.031 & 0.10 & 310.4 \\ \hline
RsR2 & 0.2658 & 0.2799 & -0.036 & 0.09 & 314.0 \\ \hline
Gil & 0.2738 & 0.2845 & -0.042 & 0.11 & 303.7 \\ \hline
\end{tabular}}

\subfloat[0.015625-0.00625 versus 0.015625 0.00625]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.2495 & 0.2519 & -0.026 & 0.06 & 392.0 \\ \hline
SdC1 & 0.2472 & 0.2519 & -0.025 & 0.05 & 400.8 \\ \hline
RdF2 & 0.2499 & 0.2508 & -0.025 & 0.05 & 404.9 \\ \hline
RsR2 & 0.2519 & 0.2500 & -0.029 & 0.04 & 402.6 \\ \hline
Gil & 0.2547 & 0.2493 & -0.026 & 0.05 & 397.1 \\ \hline
\end{tabular}}
\end{center}
\caption{Values for the Persistent Systems and Spatial Indices for all experiments with the same probability sets}
\label{fig:perspatTablesSame}
\end{figure}


\begin{figure}[t!]
\centering

\subfloat[0.99-0.4 versus 0.99-0.4]{
\includegraphics[scale = 0.75]{images/1a-plot.png}}
\quad
\subfloat[0.25-0.1 versus 0.25-0.1]{
\includegraphics[scale = 0.75]{images/2b-plot.png}}

\subfloat[0.0625-0.025 versus 0.0625-0.025]{
\includegraphics[scale = 0.75]{images/3c-plot.png}}
\quad
\subfloat[0.015625-0.00625 versus 0.015625-0.00625]{
\includegraphics[scale = 0.75]{images/4d-plot.png}}


\caption{Density against Time for all models with the same probability sets}
\label{fig:persChartSame}
\end{figure}

The simulation which was run where both species were given the same birth and death probabilities as Ruxton and Saravia's experiments (birth of 0.99 and death of 0.4) (chart \emph{a} on Figure \ref{fig:persChartSame}) showed that the population for each species for all of the discrete time update rules were roughly half the values than when there was only one species. When looking at the two species of the same update rule closely, it appears that in each case one does slightly better than the other. This becomes linearly more observable as the total population increases. It also seems as though both species interact symmetrically in such a way as to keep the total population almost constant (i.e. as the population of one species increases, the other decreases equivalently).

Interesting behaviours begin to emerge as the birth and death probabilities of the two species decreases. Although the ratio between the probabilities remain the same, the equilibrium densities for all of the different rules draw closer to population of 2500 for each species. Once the probabilities get low enough, all of the different rule settings show the same behaviour in respect to a longer time being required to reach an equilibrium and the time and final equilibrium densities being the same (as can be seen in chart \emph{d} in Figure \ref{fig:persChartSame}). The equilibrium density for each of the species for all of the settings with a low enough probability set is roughly 2500 which may relate to the fact that the ratio between the birth and death probabilities are always 2.5. The trends across the different charts in Figure \ref{fig:persChartSame} seem to show that the Gillespie simulator and the discrete step cellular automaton get more similar as the probabilities used decrease.

The densities and the Moran's I values for the experiment with both species having the same birth and death probability as Ruxton and Saravia's experiments are shown on table \emph{a} in Figure \ref{fig:perspatTablesSame}. The densities of both species all sum to be roughly equivalent to the values for the equivalent system with only one species (see Figure \ref{fig:RuxPer}). However, the Moran's I values differ greatly for some settings and are similar for others. All of the Moran's I values for the simulations with multiple species are lower than those with only one, though for SdC1 the difference is as low as 0.008 and for SdF1 it is as high as 0.1. The difference for the Gillespie simulator is 0.186 which is the highest of all the settings. The general decrease in the Moran's I is to be expected as introducing a second species increases the likelihood that the neighbours of a given cell will not be of the same type as it is.

As the probabilities drop, the densities of both species for all of the settings tend towards 2500 as shown in the plots for the densities over time (Figure \ref{fig:persChartSame}). The Moran's I values however initially all decrease and all get closer together with the values for the simulations with the lowest probabilities settings all being within 0.004 of each other (table \emph{d} on Figure \ref{fig:persChartSame}). The values for the simulation with the lowest probability set however show an increase which goes against the seeming trend from the other simulations. The values for the Gillespie simulator again become more similar to the discrete step cellular automaton as the probabilities decrease. 


%%%%%%%%%%%%%%%%%%%%%%
\subsubsection*{Spatial Indices}
\label{sec:anal.mult.spat}

For the simulations run introducing a second species to Ruxton and Saravia's experiments, the LPI were all found to have decreased (see Figure \ref{fig:perspatTablesSame}) which would be expected as the population of each species is lower allowing for a smaller number of organisms to make up the largest patch and there is also another species occupying the same space introducing competition. The number of patches for each setting has increased which shows that there are a larger number of smaller patches as opposed to one large main patch consisting of the majority of the population with the exception of a few outliers. With a smaller population size for each species plus the addition of a competing species, this would be the expected outcome.

As the probability sets get lower, the the size of the LPIs all reduce and the NPs all increase with all of the values again converging at around 0.05 and 400 for the LPIs and NPs respectively. This shows that as the probability sets decrease, the species form a larger number of very small patches and when combined with the Moran's I value being close to 0 means that they are also evenly distributed across the grid.

All of these settings show that for two species with the same probability set, the Gillespie simulator gives larger population sizes than any of the update rules for the discrete step cellular automaton for larger probabilities, however the difference reduces as the probabilities become lower until a point where there appears to be no difference.

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Species With Different Birth and Death Probabilities}
\subsubsection*{Persistent Systems}

\begin{sidewaysfigure}[h!]
\begin{center}
\subfloat[0.99-0.4 versus 0.25-0.1]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.0350 & 0.5404 & 0.077 & 0.54 & 38.0 \\ \hline
SdC1 & 0.0020 & 0.5034 & 0.103 & 0.50 & 36.2 \\ \hline
RdF2 & 0.0035 & 0.5343 & 0.100 & 0.53 & 28.3 \\ \hline
RsR2 & 0.0245 & 0.5153 & 0.067 & 0.51 & 54.9 \\ \hline
Gil & 0.8853 & 0.0012 & 0.015 & 0.89 & 6.5 \\ \hline
\end{tabular}}
\quad\quad
\subfloat[0.99-0.4 versus 0.0625-0.025]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.1249 & 0.4343 & 0.051 & 0.45 & 95.6 \\ \hline
SdC1 & 0.0027 & 0.5210 & 0.098 & 0.51 & 38.3 \\ \hline
RdF2 & 0.0064 & 0.5260 & 0.096 & 0.52 & 40.8 \\ \hline
RsR2 & 0.0824 & 0.4465 & 0.049 & 0.44 & 118.3 \\ \hline
Gil & 0.8823 & 0.0040 & 0.010 & 0.88 & 20.9 \\ \hline
\end{tabular}}

\subfloat[0.99-0.4 versus 0.015625-0.00625]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.3313 & 0.2247 & 0.010 & 0.23 & 239.8 \\ \hline
SdC1 & 0.0036 & 0.4404 & 0.136 & 0.42 & 79.4 \\ \hline
RdF2 & 0.0125 & 0.4358 & 0.135 & 0.42 & 89.5 \\ \hline
RsR2 & 0.2382 & 0.2419 & -0.007 & 0.16 & 338.0 \\ \hline
Gil & 0.8705 & 0.0157 & -0.010 & 0.87 & 79.0 \\ \hline
\end{tabular}}
\quad\quad
\subfloat[0.25-0.1 versus 0.0625-0.025]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.3224 & 0.2426 & -0.024 & 0.18 & 258.2 \\ \hline
SdC1 & 0.1796 & 0.3457 & 0.013 & 0.32 & 206.0 \\ \hline
RdF2 & 0.2043 & 0.3379 & 0.000 & 0.29 & 218.6 \\ \hline
RsR2 & 0.3036 & 0.2406 & -0.028 & 0.15 & 280.6 \\ \hline
Gil & 0.5105 & 0.0866 & -0.001 & 0.51 & 164.8 \\ \hline
\end{tabular}}

\subfloat[0.25-0.1 versus 0.015625-0.00625]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.4034 & 0.1578 & -0.026 & 0.323 & 300.1 \\ \hline
SdC1 & 0.3011 & 0.2073 & -0.008 & 0.100 & 338.8 \\ \hline
RdF2 & 0.3339 & 0.1971 & -0.017 & 0.126 & 321.9 \\ \hline
RsR2 & 0.3812 & 0.1581 & -0.032 & 0.270 & 320.9 \\ \hline
Gil & 0.4927 & 0.1031 & -0.027 & 0.486 & 254.4 \\ \hline
\end{tabular}}
\quad\quad
\subfloat[0.0625-0.025 versus 0.015625-0.00625]{
\begin{tabular}{|c|c|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} & \multirow{2}{*}{\textbf{LPI}} & \multirow{2}{*}{\textbf{NP}}\\ \cline{2-3}
& 1 & 2 & & &  \\ \hline
SdF1 & 0.2543 & 0.1857 & -0.034 & 0.22 & 340.4 \\ \hline
SdC1 & 0.3449 & 0.1846 & -0.032 & 0.017 & 348.5 \\ \hline
RdF2 & 0.3475 & 0.1865 & -0.033 & 0.019 & 339.7 \\ \hline
RsR2 & 0.3506 & 0.1858 & -0.036 & 0.22 & 332.3 \\ \hline
Gil & 0.3815 & 0.1663 & -0.036 & 0.30 & 332.3 \\ \hline
\end{tabular}}

\end{center}
\caption{Values for the Persistent Systems and Spatial Indices experiments for all experiments with the differing probability sets}
\label{fig:perspatTablesDiff}
\end{sidewaysfigure}

\begin{sidewaysfigure}[h!]
\centering

\subfloat[0.99-0.4 versus 0.25-0.1]{
\includegraphics[scale = 0.75]{images/1b-plot.png}}
\quad
\subfloat[0.99-0.4 versus 0.0625-0.025]{
\includegraphics[scale = 0.75]{images/1c-plot.png}}

\subfloat[0.99-0.4 versus 0.015625-0.00625]{
\includegraphics[scale = 0.75]{images/1d-plot.png}}
\quad
\subfloat[0.25-0.1 versus 0.0625-0.025]{
\includegraphics[scale = 0.75]{images/2c-plot.png}}

\subfloat[0.25-0.1 versus 0.015625-0.00625]{
\includegraphics[scale = 0.75]{images/2d-plot.png}}
\quad
\subfloat[0.0625-0.025 versus 0.015625-0.00625]{
\includegraphics[scale = 0.75]{images/3d-plot.png}}

\caption{Density against Time for all models with the differing probability sets}
\label{fig:persChartDiff}
\end{sidewaysfigure}

From the charts in Figure \ref{fig:persChartDiff} it is clear that as the probability sets get lower, the systems take longer to stabilise to the point that even within the 1500 steps, most of the species in charts \emph{c} to \emph{f} do not establish an equilibrium density. From the charts it is still possible to see that for each simulation, one species ``wins" (one survives while the other is driven to extinction). In each of the rules for the cellular automaton, it is always the second species (the one with the lower birth and death probabilities) that wins, though as the probabilities get lower, the charts in Figure \ref{fig:persChartDiff} show that initially the species with the higher probabilities do better until they reach a ``peak" density (generally around 0.4) at which point it starts to decrease until it reaches extinction as the species with the lower probabilities rises to an equilibrium density. For the Gillespie simulator however,  it is always the species with the higher probabilities which wins. The charts also show that although the Gillespie simulator shows the same trends as the cellular automaton where lower probabilities results
 in a longer time to stabilise, it does so quicker than any of the discrete step update rules. When all of the simulations are allowed to run long enough to allow the system to stabilise, the final populations of all the discrete step cellular automaton are between 0.55 and 0.57 and for the Gillespie simulator, the values tend towards this same point as the probabilities of the winning species decreases. This suggests that the cellular automaton will result in similar systems as long as the ratios of the birth and death probabilities remain constant, though it will take longer to stabilise if the probabilities are lower. However the Gillespie simulator is sensitive to the actual values chosen for the probabilities and not just the ratio between them, though as the probabilities get smaller, it begins to show more similarities to the cellular automaton.

The density and Moran's I values presented in Figure \ref{fig:perspatTablesDiff} do not show much usable data as the values are averaged over every step in the simulation. This means that for the simulations which did not stabilise, or for those that only stabilised just before the end of the simulation, the values will not be indicative of the stabilised system. The results for the simulations presented in charts \emph{e} and \emph{f} run for over 100,000 steps without many of the rules reaching an equilibrium though show clearly show the same trends as the other simulations. When the simulations are allowed to run until they stabilise and only the last 200 values are taken (all from when the system is stable), all of the discrete step update rules show very similar values for the density and the Moran's I values with the results only deviating by 0.002 and 0.004 respectively from Figure \ref{fig:stabSysCA}. The Moran's I values in these figures are equivalent for those with only one species since the other is extinct by this point.

\begin{figure}[H]
\centering
\begin{tabular}{|c|c|c|c|}\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} \\ \cline{2-3}
& 1 & 2 &  \\ \hline
SdF1 & 0.0 & 0.5665 & 0.084 \\ \hline
SdC1 & 0.0 & 0.5643 & 0.067 \\ \hline
RdF2 & 0.0 & 0.5675 & 0.076 \\ \hline
RsR2 & 0.0 & 0.5668 & 0.074 \\ \hline
\end{tabular}
\caption{Averages for the last 200 figures for the simulations for every probability set combination from Figure \ref{fig:persChartDiff} run until they stabilised}
\label{fig:stabSysCA}
\end{figure}

The Gillespie simulator however showed more sensitivity to lowering probabilities as shown in Figure \ref{fig:stabSysGil}

\begin{figure}[H]
\centering
\begin{tabular}{|c|c|c|c|}\hline
\textbf{Probability set of} & \multicolumn{2}{c|}{\textbf{Density}} & \multirow{2}{*}{\textbf{Moran's I}} \\ \cline{2-3}
\textbf{winning species} & 1 & 2 &  \\ \hline
0.99-0.4 & 0.8853 & 0.0 & 0.022 \\ \hline
0.25-0.1 & 0.6014 & 0.0 & 0.076 \\ \hline
0.0625-0.025 & 0.5693 & 0.0 & 0.0785 \\ \hline
0.015625-0.00625 & 0.5659 & 0.0 & 0.0901 \\ \hline
\end{tabular}
\caption{Averages for the last 200 figures for each probability set with the Gillespie simulator run until they stabilised}
\label{fig:stabSysGil}
\end{figure}

These values again show that the outcome of the cellular automaton is only influenced by the ratio of the birth and death probability and the actual values are only determining how long the system takes to stabilise, whereas the Gillespie simulator is sensitive to the actual birth and death probabilities. It also shows that the Gillespie simulator becomes more similar to the cellular automaton as the birth and death probabilities get smaller although it is still the species with the higher probabilities and not the lower ones that wins.

\subsubsection*{Spatial Indices}
For the same reasons as mentioned above, the values for the LPI and NP shown in the tables in Figure \ref{fig:perspatTablesDiff} are not indicative of the actual systems. I determined more accurate results using the same methods as before and again found that the different update rules for the cellular automaton all returned very similar values for the LPI and NP when allowed to stabilise regardless of the probabilities. The values for the LPI and NP only deviated by at most 0.04 and 5 respectively for the different probability settings shown in Figure \ref{fig:stabSysSpatCA}

\begin{figure}[H]
\centering
\begin{tabular}{|c|c|c|}\hline
\textbf{Model} & \textbf{LPI} & \textbf{NP} \\ \hline
SdF1 & 0.5631 & 21 \\ \hline
SdC1 & 0.5496 & 18 \\ \hline
RdF2 & 0.5637 & 8 \\ \hline
RsR2 & 0.5588 & 16 \\ \hline
\end{tabular}
\caption{Averages for the last 200 figures for the simulations for every probability set combination from Figure \ref{fig:persChartDiff} run until they stabilised}
\label{fig:stabSysSpatCA}
\end{figure}

The Gillespie simulator again returned varying values for the LPI and NP as the birth and death probabilities changed as shown in Figure \ref{fig:stabSysSpatGil}. This table shows that the species for Gillespie simulator begins to reduce in population size and begins to have more patches separate from the main one, however the largest patch still encompasses most of the population.

\begin{figure}[H]
\centering
\begin{tabular}{|c|c|c|}\hline
\textbf{Model} & \textbf{LPI} & \textbf{NP} \\ \hline
0.99-0.4 & 0.8851 & 1 \\ \hline
0.25-0.1 & 0.6001 & 7 \\ \hline
0.0625-0.025 & 0.5591 & 17 \\ \hline
0.015625-0.00625 & 0.5593 & 20 \\ \hline
\end{tabular}
\caption{Averages for the last 200 figures for each setting with the Gillespie simulator run until they stabilised}
\label{fig:stabSysSpatGil}
\end{figure}

\section{Analysis of Experiments}
The experiments which were run were able to highlight certain key differences between a Gillespie simulator and a discrete step cellular automaton in regards to the population size some spatial behaviour under certain circumstances. The Extinction Medians experiment was an effective way of determining the ``strength" of the species for the different update rules as well as the Gillespie simulator, however the probabilities chosen did not lead to an extinction for any of the runs for the Gillespie simulator as the birth probability was too high. To improve this, the experiments could be run either using an even higher death probability, though that may lead to indistinguishable values for the weaker update rules, or by choosing a different set of birth and death probabilities which still have the same ratio except with smaller values.

 For the Persistent Systems experiments, the values obtained were not always indicative of how the simulation unfolded as it assumed that all of the runs were in a stabilised state for the majority of the run. In order to be truly comparable, all rules for each simulation would have to have been stable for the same period of time. This could have been better measured if the values were only averaged for the last \emph{X} steps in the simulation where \emph{X} is less than or equal to the number of steps for which the system has been stable for all update rules. Moran's I may not have been the best choice as a value for determining the spatial behaviour for systems with multiple species as it was not designed to take into account two completely separate values on the grid which have no correlation to each other, but was rather intended for two distinct states (black or white) and different values inbetween the two (differing shades of grey). The same problems in regards to the stability of the simulation also applies to the Spatial Indices experiments and can also be fixed in the same way.

The discrete step update rules were ones randomly chosen by Ruxton and Saravia and all of the rules (with the exception of RsR2) evaluated for death first, meaning that the systems would have been weaker than if they had been evaluated for birth first. This difference would most likely have given the Gillespie simulator an advantage as it was not restricted by this setting. In order for a fairer evaluation, the experiments could either have been run with the chosen rules with both birth-first and death-first or a new set of randomly selected update rules could have been chosen since Ruxton and Saravia had not considered birth-first options.

The probability sets chosen for the experiment are all of the same ratio, starting with one chosen by Ruxton and Saravia with no reasoning for the decision given. The extremely high birth probability is one which is unlikely to be used for any ecological modelling and as such, it could be argued that the results obtained from using it do not have any real significance. Since all of the other sets were directly derived from the first one in a manner that kept the ratio of the birth to death probability the same for every simulation, the experiments were not as comprehensive as they could have been had more varying probability sets been used.

\subsection*{Summary}
This chapter showed that there are significant differences in the outcomes of the models between the Gillespie simulator and the cellular automata. An analysis of the experiments were also presented showing that some improvements could be made for future work and that there are further sections, such as further varying the probability sets, which should be investigated further. It also showed that as the probability sets reduced, the different update rules for the cellular automata did not in fact have much of an impact for the outcome of the models.
